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1、GPS时间 t
作为统一的时间基准，保证卫星和接收机时钟的同步
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                                <h2>
                                    2.5 伪距定位与多普勒测速
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                                    2023-10-12, 699 words, 4 min read
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                                        <h2 id="一-相关时间定义">一、相关时间定义</h2>
<h4 id="1-gps时间-t">1、GPS时间 t</h4>
<p>作为统一的时间基准，保证卫星和接收机时钟的同步</p>
<h4 id="2-gps信号发射时间-ts">2、GPS信号发射时间 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>t</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">t^{(s)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">s</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></h4>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>t</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>t</mi><mo>+</mo><mi>δ</mi><msup><mi>t</mi><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">t^{(s)}(t)=t+\delta t^{(S)}(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">s</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69841em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.05764em;">S</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span>   其中 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><msup><mi>t</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\delta t^{(s)}(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">s</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span> 为误差项</p>
<h4 id="3-gps信号接收时间-t_u">3、GPS信号接收时间 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>t</mi><mi>u</mi></msub></mrow><annotation encoding="application/x-tex">t_u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76508em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></h4>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>t</mi><mi>u</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>t</mi><mo>+</mo><mi>δ</mi><msub><mi>t</mi><mi>u</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">t_u(t)=t+\delta t_u(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69841em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span>  其中 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><msub><mi>t</mi><mi>u</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\delta t_u(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span> 为误差项</p>
<h2 id="二-伪距">二、伪距</h2>
<h4 id="1-伪距定义">1、伪距定义</h4>
<p>假设信号从卫星到接收机所需要的实际传播时间为 $\tau $</p>
<p>则伪距定义式   <img src="http://cos.pansis.site/202310121654475.png/abc123" alt="image-20231012165400276" style="zoom:33%;" /></p>
<h4 id="2-伪距观测方程">2、伪距观测方程</h4>
<img src="http://cos.pansis.site/202310121654841.png/abc123" alt="image-20231012165457783" style="zoom:33%;" />
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>I</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">I(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span> 表示电离层延时（数学模型推导的已知量）</p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>T</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">T(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span> 表示对流层延时（数学模型推导的已知量）</p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><msup><mi>t</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\delta t^{(s)}(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">s</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span> 可以通过第一个数据块中的时钟校正参数精准计算得到</p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>r</mi><mo>(</mo><mi>t</mi><mo>−</mo><mi>τ</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">r(t-\tau,t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.1132em;">τ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span> 表示<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(t-\tau)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.1132em;">τ</span><span class="mclose">)</span></span></span></span> 时刻卫星位置与 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span> 时刻接收机位置之间的直线距离（实际需要计算的量）</p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ρ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\rho(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ρ</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span> 表示伪距，伪距通过伪距码测得的距离（不是直线距离）</p>
<p>推导：</p>
<figure data-type="image" tabindex="1"><img src="http://cos.pansis.site/202310121656088.png/abc123" alt="image-20231012165611932" loading="lazy"></figure>
<h2 id="三-卫星位置求解">三、卫星位置求解</h2>
<p>卫星位置  <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msup><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo separator="true">,</mo><msup><mi>y</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo separator="true">,</mo><msup><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">(x^{(n)},y^{(n)},z^{(n)})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p>
<h4 id="1-求解信号发射时间-t_s">1、求解信号发射时间 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>t</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">t_{s}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76508em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></h4>
<p>公式：<img src="http://cos.pansis.site/202310121707002.png/abc123" alt="image-20231012170758922" style="zoom:33%;" /></p>
<img src="http://cos.pansis.site/202310121708154.png/abc123" alt="image-20231012170818031" style="zoom:33%;" />
<p>TOW时刻存在于导航电文中，是接收器可确定的卫星时间</p>
<h4 id="2-求解卫星位置">2、求解卫星位置</h4>
<p>1、求解<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(t-\tau )</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.1132em;">τ</span><span class="mclose">)</span></span></span></span>时刻（信号发射时刻）卫星位置 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>x</mi><mi>k</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>k</mi></msub><mo separator="true">,</mo><msub><mi>z</mi><mi>k</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">(x_k,y_k,z_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p>
<p><img src="http://cos.pansis.site/202310121714495.png/abc123" alt="image-20231012171423204" style="zoom:15%;" /><img src="http://cos.pansis.site/202310121714796.png/abc123" alt="image-20231012171444523" style="zoom:15%;" /><img src="http://cos.pansis.site/202310121715245.png/abc123" alt="image-20231012171507992" style="zoom:15%;" /></p>
<p>2、求解<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>时刻（信号接收时刻）卫星位置 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msup><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo separator="true">,</mo><msup><mi>y</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo separator="true">,</mo><msup><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">(x^{(n)},y^{(n)},z^{(n)})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p>
<img src="http://cos.pansis.site/202310121715330.png/abc123" alt="image-20231012171518256" style="zoom:50%;" />
<h4 id="四-卫星钟差求解">四、卫星钟差求解</h4>
<p>卫星钟差 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><msup><mi>t</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\delta t^{(s)}(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">s</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span></p>
<p>公式：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><msup><mi>t</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msup><mo>=</mo><mi mathvariant="normal">△</mi><mi>δ</mi><msup><mi>t</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msup><mo>+</mo><mi mathvariant="normal">△</mi><msub><mi>t</mi><mi>r</mi></msub><mo>−</mo><msub><mi>T</mi><mrow><mi>G</mi><mi>D</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\delta t^{(s)}=\triangle  \delta t^{(s)}+\triangle t_r-T_{GD}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">s</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0824399999999998em;vertical-align:-0.19444em;"></span><span class="mord">△</span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">s</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">△</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">G</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<img src="http://cos.pansis.site/202310121718706.png/abc123" alt="image-20231012171828536" style="zoom:33%;" />
<h2 id="五-伪距定位算法">五、伪距定位算法</h2>
<h4 id="1-已知数据和初始解">1、已知数据和初始解</h4>
<img src="http://cos.pansis.site/202310121719003.png/abc123" alt="image-20231012171941937" style="zoom:33%;" />
<h4 id="2-非线性方程线性化">2、非线性方程线性化</h4>
<p>伪距方程去除常数项后的方程为</p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msqrt><mrow><mo>(</mo><msup><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>−</mo><mi>x</mi><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><msup><mi>y</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>−</mo><mi>y</mi><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><msup><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>−</mo><mi>z</mi><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt><mo>+</mo><mi>c</mi><mi>δ</mi><msub><mi>t</mi><mi>u</mi></msub><mo>=</mo><msup><mi>ρ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\sqrt{(x^{(n)}-x)^2+(y^{(n)}-y)^2+(z^{(n)}-z)^2} +c\delta t_u=\rho^{(n)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.2729999999999999em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670000000000001em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.814em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.814em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.814em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.9270000000000005em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg width='400em' height='1.28em' viewBox='0 0 400000 1296' preserveAspectRatio='xMinYMin slice'><path d='M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167,
158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067
c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84,
175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71
c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3,
-252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26
s76,-59,76,-59s76,-60,76,-60z M1001 80H40000v40H1012z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2729999999999999em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0824399999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">ρ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>求导经过线性化得</p>
<p><img src="http://cos.pansis.site/202310121724275.png/abc123" alt="image-20231012172422222" style="zoom:50%;" />                       <img src="http://cos.pansis.site/202310121724684.png/abc123" alt="image-20231012172402479" style="zoom:15%;" /></p>
<h4 id="3-求解线性方程">3、求解线性方程</h4>
<p>最小二乘法</p>
<img src="http://cos.pansis.site/202310121725134.png/abc123" alt="image-20231012172513088" style="zoom:33%;" />
<h4 id="4-更新线性方程组的根">4、更新线性方程组的根</h4>
<p>梯度下降</p>
<img src="http://cos.pansis.site/202310121725372.png/abc123" alt="image-20231012172539308" style="zoom:33%;" />
<h4 id="5-判断收敛性">5、判断收敛性</h4>
<img src="http://cos.pansis.site/202310121725472.png/abc123" alt="image-20231012172556429" style="zoom:33%;" />
<h2 id="六-多普勒测速">六、多普勒测速</h2>
<h4 id="1-伪距速度方程">1、伪距速度方程</h4>
<img src="http://cos.pansis.site/202310121727918.png/abc123" alt="image-20231012172713846" style="zoom:33%;" />
<p>与测距方法完全不同，利用多普勒频移的测量原理，误差比伪距的测量误差低1至2个数量级</p>
<h4 id="2-求解">2、求解</h4>
<p>与伪距测量类似</p>
<img src="http://cos.pansis.site/202310121728853.png/abc123" alt="image-20231012172825610" style="zoom:33%;" />
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